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Outcomes of interest

A total of 1000 datasets were generated for each of the 24 settings. The generated data were subsequently analyzed by fitting Model (15.1) in SAS. Two different parameterizations for the D matrix were considered. First, a completely general (unstructured; UN) D matrix that is parameterized directly in terms of variances and covariances. Second, a non-diagonal factor-analytic structure with 4 factors (FA0(4)). The latter structure specifies a Cholesky root parameterization for the 4 x 4 unstructured blocks in D. This leads to a substantial simplification of the optimization problem, i.e., the problem now changes from a constrained one to an unconstrained one. The FA0(4) structure has f (2# — q + 1) covariance parameters, where q refers to the number of factors and t is the dimension of the matrix. In the present setting, the FA0(4) structure thus has a total of 10 parameters. These parameters are used to compute the components in D, i.e., the (i, j)th element of D is computed as . The Cholesky root parameterization ensures that

D (and Vi) is positive-definite during the entire estimation process (West et al., 2007).

The key outcome of interest in the simulations was model convergence. Three model convergence categories were distinguished: (i) proper convergence, i.e., the model converged and the variance-covariance matrix of the random effects (D) and the final Hessian (H), used to compute the standard errors of the covariance parameters, were positive-definite (PD); (ii) the model converged but D or H was not PD; and finally, (iii) divergence.

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